Atticus and Alexandra disagree about how to deal with the children. How does Atticus handle the situation?

In the novel, there are two major disagreements between Alexandra and Atticus: raising children and race.

When it comes to raising children, Alexandra is much more traditional and conservative (conforming to the norms in society). In particular, she wants Scout to act like a lady. As it stands, Scout is very much a tomboy. This is why Alexandra tries to make Scout wear a dress. She should act like a proper lady and look like...

In the novel, there are two major disagreements between Alexandra and Atticus: raising children and race.

When it comes to raising children, Alexandra is much more traditional and conservative (conforming to the norms in society). In particular, she wants Scout to act like a lady. As it stands, Scout is very much a tomboy. This is why Alexandra tries to make Scout wear a dress. She should act like a proper lady and look like one. Atticus does not concern himself with these things. Instead, he wants Scout and Jem to be children. So, he does not impose social constraints on them. The very fact that his children call him Atticus is telling.

Second, Atticus does not believe in shielding his children from the ugly racism in the world. To a degree he does, but by the end of the book, he allows both Jem and Scout not only to know but also experience the racism in Maycomb. Alexandra would very much like to protect the children. For example, when Atticus talks about race freely in the house, Alexandra is shocked. She says:

"Don't talk like that in front of them... Like that in front of Calpurnia. You said Braxton Underwood despises Negroes right in front of her."

I think that you are talking about what happens at the end of Chapter 13. In that passage, Atticus comes to talk to Jem and Scout. It is clear to them that Aunt Alexandra has asked him to do this.

What she wants the kids to do is to be more aware of their place in the social structure of Maycomb. She wants them to act more as if they are important because they come from a good family. Atticus tries to deliver the message, but you can tell his heart isn't in it. He eventually just tells them to forget it -- they should go back to acting how they usually do.

Atticus takes the situation into his own hands and tells Jem and Scout about how their family is highly superior to the rest in Maycomb and Scout and Jem are shocked at the same time a little angered by this